To calculate the expected frequency for each value of $x$, we need to determine the expected probability of getting 4, 5, or 6 on a die in a single throw, and then use that to find the expected number of times each outcome occurs in 96 throws.

The values given appear to be the number of times that 4, 5, or 6 were actually thrown. We also need to calculate the probability of rolling 4, 5, or 6 on a single die.

Step 1: Probability Calculation

On a fair die, there are 6 faces (1 through 6), and each face has an equal chance of being rolled. The probability of rolling either a 4, 5, or 6 in a single throw of one die is:

$P(\text{4, 5, or 6}) = \frac{3}{6} = \frac{1}{2}$

Thus, the probability of rolling a 4, 5, or 6 on a single die is $\frac{1}{2}$.

Step 2: Expected Frequency Calculation

Now, let’s calculate the expected frequency of throwing 4, 5, or 6 based on 96 trials.

For each number of dice showing a 4, 5, or 6 (denoted by $x$), the expected frequency $E(f_x)$ is given by:

$E(f_x) = \text{total number of trials} \times P(\text{4, 5, or 6 on a die}) = 96 \times P(\text{4, 5, or 6}) = 96 \times \frac{1}{2} = 48$

Step 3: Interpreting the Data

The frequency distribution $f(x)$ is given as follows:

$x$	0	1	2	3	4	5
$f(x)$	1	10	24	35	18	8

• Total number of occurrences: The sum of all $f(x)$ values should be equal to the total number of trials (96). Let's check:

$1 + 10 + 24 + 35 + 18 + 8 = 96$

Thus, the observed data matches the total number of trials.

Step 4: Calculating Expected Frequency for Each $x$

The expected frequency for each value of $x$ (number of dice showing 4, 5, or 6) can be calculated using the binomial distribution, since each die throw is independent.

The number of successes (dice showing 4, 5, or 6) follows a binomial distribution with:

• $n = 5$ (the number of dice)
• $p = \frac{1}{2}$ (the probability of success on each die)

For a binomial distribution, the expected frequency of exactly $x$ successes is:

$E(f_x) = \binom{5}{x} \times \left(\frac{1}{2}\right)^x \times \left(\frac{1}{2}\right)^{5-x} \times 96 = \binom{5}{x} \times \left(\frac{1}{2}\right)^5 \times 96$

Step 5: Computing the Expected Frequency for Each Value of $x$

Now, we compute the expected frequency for each $x$ from 0 to 5:

1. For $x = 0$: $E(f_0) = \binom{5}{0} \times \left(\frac{1}{2}\right)^5 \times 96 = 1 \times \frac{1}{32} \times 96 = 3$

2. For $x = 1$: $E(f_1) = \binom{5}{1} \times \left(\frac{1}{2}\right)^5 \times 96 = 5 \times \frac{1}{32} \times 96 = 15$

3. For $x = 2$: $E(f_2) = \binom{5}{2} \times \left(\frac{1}{2}\right)^5 \times 96 = 10 \times \frac{1}{32} \times 96 = 30$

4. For $x = 3$: $E(f_3) = \binom{5}{3} \times \left(\frac{1}{2}\right)^5 \times 96 = 10 \times \frac{1}{32} \times 96 = 30$

5. For $x = 4$: $E(f_4) = \binom{5}{4} \times \left(\frac{1}{2}\right)^5 \times 96 = 5 \times \frac{1}{32} \times 96 = 15$

6. For $x = 5$: $E(f_5) = \binom{5}{5} \times \left(\frac{1}{2}\right)^5 \times 96 = 1 \times \frac{1}{32} \times 96 = 3$

Step 6: Conclusion

The expected frequency for each value of $x$ is:

$x$	0	1	2	3	4	5
$E(f_x)$	3	15	30	30	15	3

These are the expected frequencies of throwing 4, 5, or 6 on five dice across 96 throws.

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